[seqfan] Pythagorean areas and congrua

israel at math.ubc.ca israel at math.ubc.ca
Sat Apr 4 02:19:23 CEST 2015


A073120 has the Name "Areas of right triangles with integer sides". But it 
doesn't contain the areas of all Pythagorean triangles, just those with 
sides of the form (2mn, m^2-n^2, m^2+n^2). For example, the sequence does 
not contain 54, the area of the Pythagorean triangle with sides (9,12,15). 
The sequence with the areas of all Pythagorean triangles is A009112.

A057102 has the Name "Congrua (possible solutions to the congruum problem): 
numbers n such that there are integers x, y and z with n = x^2-y^2 = 
z^2-x^2." Now n is a "congruum" iff n/4 is the area of a Pythagorean 
triangle. So we should have A057102(n) = 4*A009112(n). For example, 4*54 = 
216 should be in the sequence: 216 = 15^2 - 3^2 = 21^2 - 15^2. 
Unfortunately, it's not in the Data: the Formula for A057102 is a(n) = 4 * 
A073120(n) and the Data seem to match that.

The sequence 4*A009112(n) does not seem to be in the OEIS.  It should start
24, 96, 120, 216, 240, 336, 384, 480, 600, 720, 840, 864, 960.

This has propagated to Wikipedia: http://en.wikipedia.org/wiki/Congruum says

The first few congrua are:

    24, 96, 120, 240, 336, 384, 480, 720, 840, 960 ¨ (sequence A057102 in 
OEIS).

and similarly MathWorld: http://mathworld.wolfram.com/Congruum.html

So, what to do?

Cheers,
Robert Israel



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